On the exponent in the "generalized" matching equation.
The curved matching equation is mathematically proven, so you can safely use it to predict choice in clinic and classroom.
01Research in Context
What this study did
Allen (1981) wrote a math proof. The paper shows the power-function matching equation is algebraically sound.
No people or animals were tested. The work is pure theory.
What they found
The proof says the generalized matching law is valid when reinforcement ratios stay in a fixed form.
In plain words, the curved version of the matching equation is not a statistical trick. It is real math.
How this fits with other research
Michael (1974) laid the ground rules seven years earlier. That paper listed the formal rules for basic matching. Allen (1981) adds the proof for the curved, power version.
Oliver et al. (2002) later tested kids with severe problem behavior. Response ratios matched reinforcement ratios, just as the math predicts. The lab proof now supports clinic data.
Thorne (2010) throws a curve. That paper claims bias and sensitivity terms are artifacts of sloppy models. Allen (1981) legitimizes those same terms. The clash is friendly: one side says the terms are real math, the other says they can vanish with better model specs.
Why it matters
You can now trust the power-function matching equation when you see curved data. Use it to predict how choice will shift after you change reinforcement rates in token boards, break schedules, or functional communication training. If the curve looks flat or steep, the equation still holds; you are not mis-reading noise.
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02At a glance
03Original abstract
A power function equation between ratios of behavior and ratios of reinforcement rates has been called a generalized form of Herrnstein's (1961) matching law, even without a formal relationship having been shown between the two equations. The present work uses a functional relationship to prove that when ratios of reinforcement are not equivalent to ratios of behavior, and the transform leading to this inequality is consistent for every pair of reinforcement rates, the result is a power function relationship between response and reinforcement ratios. The label "generalized matching equation" for the power function equation is thus validated formally.
Journal of the experimental analysis of behavior, 1981 · doi:10.1901/jeab.1981.35-125